Help constructing an OBB, trying to Represent a matrix with 3 vectors

Question

I am presently trying to construct an OBB (Oriented Bounding Box) using the source and math contained in the book "Real Time Collision Detection".

One problem with the code contained in this book is that it does very little to explain what the parameters mean for the methods.

I am trying to figure out what I need to feed my setOBB() method (I wrote this one). It goes like this:

void PhysicalObject::setOBB( Ogre::Vector3 centrePoint, Ogre::Vector3 localAxes[3], Ogre::Vector3 positiveHalfwidthExtents )
{
    // Ogre::Vector3 c;      // OBB center point
    // Ogre::Vector3 u[3]; // Local x-, y-, and z-axes (rotation matrix)
    // Ogre::Vector3 e; // Positive halfwidth extents of OBB along each axis

    m_obb.c         = centrePoint;
    m_obb.u[0]      = localAxes[0];
    m_obb.u[1]      = localAxes[1];
    m_obb.u[2]      = localAxes[2];
    m_obb.e         = positiveHalfwidthExtents;
}

Looking at the parameters it wants above, the first and third parameters I believe I understand.

1. Pass in the centre position of the object.
2. This is my problem. I believe it wants a matrix represented using an array of 3 vectors? but how?
3. A Vector which contains magnitude for the distance between the centre point and the edge of the OBB in each x,y,z direction.

Here is what I'm doing currently:

// Build the OBB
Ogre::Vector3 rotation[3];
Ogre::Vector3 centrePoint       = sphere->getPosition();
rotation[0]      = ?
rotation[1]      = ?
rotation[2]      = ?
Ogre::Vector3 halfEdgeLengths   = Ogre::Vector3( 1,1,1 );

myObject->setOBB( centrepoint, rotation, halfEdgeLengths );

How can i represent a matrix using three vectors (which I cannot avoid doing this way). Thanks.

Answer 1

Ok. So another question is that this functionality requires it to be a matrix, right? I'm assuming so, or you'd just use the individual axes vectors in the places they are needed during the calculation.

That said, if you stuff these vectors into a matrix in a uniform way, then you'll either have them as the rows of the matrix or the columns of the matrix and, more or less, have a 50% chance to get it right depending on how the functions accepting the matrix expect the matrix to be formatted. So, set up the 2d array as the matrix and go forward with caution, I'd suggest. For example:

int matrix[3][3] = {     { x1, x2, x3 },
                         { y1, y2, y3 },
                         { z1, z2, z3 }
                   };

Flip if you need to go column-major.

Answer 2

A 3x3 matrix representing a rotation/scale in 3d space is nothing more than three vectors in row. Each column vector is the rotated and scaled main axis. First column is the scaled and rotated x axis, second and third are y and z. (Ogre uses column major matrices)

So, localAxes[3] simply is a rotation. And you can get it from a Quaternion.

Ogre::Vector3 rotation[3];
Ogre::Quaternion orientation = sphere->getOrientation();
orientation.ToAxes(rotation);
// Now rotation has the three axes representing the orientation of the sphere.